A note on multicolour Ramsey numbers and random sphere graphs

TL;DR

This paper improves lower bounds on multicolour Ramsey numbers by replacing binomial random graphs with random sphere graphs, leveraging recent advances in geometric graph models.

Contribution

It introduces an exponential improvement in lower bounds for multicolour Ramsey numbers using random sphere graphs instead of binomial graphs.

Findings

Exponential lower bound improvement for $r(t; ext{ell})$

Application of random sphere graphs in Ramsey theory

Enhanced understanding of geometric graph models in combinatorics

Abstract

The Ramsey number $r(t;\ell)$ is the smallest $n$ such that every $\ell$-coloring of the edges of $K_n$ gives a monochromatic $K_{t}$. In recent years, there have been several improvements on asymptotic lower bounds for these numbers when $\ell\geq 3$. This started with a breakthrough result of Conlon and Ferber, followed by further improvements of Wigderson and then Sawin. Building on the previous approaches, Sawin used blowups of an unbalanced binomial random graph to show that there is some explicit constant $\delta_*\approx 0.383796$ such that $r(t;\ell)\geq 2^{\delta_*(\ell-2)t+t/2+o(t)}$. In this short note, we show that one can get an exponential improvement in this bound by replacing the use of a binomial random graph with a random sphere graph, a model which which has recently been applied by Ma, Shen and Xie in a breakthrough on lower bounds for (2-colour) Ramsey numbers in the (slightly) off-diagonal setting.